Studying genetic regulatory networks at the molecular level: Delayed reaction stochastic models

Bye bye, linearity, bye: quantification of the mean for linear CRNs in a random environment

Molecular reactions within a cell are inherently stochastic, and cells often differ in morphological properties or interact with a heterogeneous environment. Consequently, cell populations exhibit heterogeneity both due to these intrinsic and extrinsic causes. Although state-of-the-art studies that focus on dissecting this heterogeneity use single-cell measurements, the bulk data that shows only the mean expression levels is still in routine use. The fingerprint of the heterogeneity is present also in bulk data, despite being hidden from direct measurement. In particular, this heterogeneity can affect the mean expression levels via bimolecular interactions with low-abundant environment species. We make this statement rigorous for the class of linear reaction systems that are embedded in a discrete state Markov environment. The analytic expression that we provide for the stationary mean depends on the reaction rate constants of the linear subsystem, as well as the generator and stationary distribution of the Markov environment. We demonstrate the effect of the environment on the stationary mean. Namely, we show how the heterogeneous case deviates from the quasi-steady state (Q.SS) case when the embedded system is fast compared to the environment.

What Is a Digital Twin? Experimental Design for a Data-Centric Machine Learning Perspective in Health

The idea of a digital twin has recently gained widespread attention. While, so far, it has been used predominantly for problems in engineering and manufacturing, it is believed that a digital twin also holds great promise for applications in medicine and health. However, a problem that severely hampers progress in these fields is the lack of a solid definition of the concept behind a digital twin that would be directly amenable for such big data-driven fields requiring a statistical data analysis. In this paper, we address this problem. We will see that the term 'digital twin', as used in the literature, is like a Matryoshka doll. For this reason, we unstack the concept via a data-centric machine learning perspective, allowing us to define its main components. As a consequence, we suggest to use the term Digital Twin System instead of digital twin because this highlights its complex interconnected substructure. In addition, we address ethical concerns that result from treatment suggestions for patients based on simulated data and a possible lack of explainability of the underling models.

Bacterial degrons in synthetic circuits

Bacterial proteases are a promising post-translational regulation strategy in synthetic circuits because they recognize specific amino acid degradation tags (degrons) that can be fine-tuned to modulate the degradation levels of tagged proteins. For this reason, recent efforts have been made in the search for new degrons. Here we review the up-to-date applications of degradation tags for circuit engineering in bacteria. In particular, we pay special attention to the effects of degradation bottlenecks in synthetic oscillators and introduce mathematical approaches to study queueing that enable the quantitative modelling of proteolytic queues.

Mechanistic models of blood cell fate decisions in the era of single-cell data

Billions of functionally distinct blood cells emerge from a pool of hematopoietic stem cells in our bodies every day. This progressive differentiation process is hierarchically structured and remarkably robust. We provide an introductory review to mathematical approaches addressing the functional aspects of how lineage choice is potentially implemented on a molecular level. Emerging from studies on the mutual repression of key transcription factors, we illustrate how those simple concepts have been challenged in recent years and subsequently extended. Especially, the analysis of omics data on the single-cell level with computational tools provides descriptive insights on a yet unknown level, while their embedding into a consistent mechanistic and mathematical framework is still incomplete.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

Insights into archaeal chaperone machinery: a network-based approach

Molecular chaperones are a diverse group of proteins that ensure proteome integrity by helping the proteins fold correctly and maintain their native state, thus preventing their misfolding and subsequent aggregation. The chaperone machinery of archaeal organisms has been thought to closely resemble that found in humans, at least in terms of constituent players. Very few studies have been ventured into system-level analysis of chaperones and their functioning in archaeal cells. In this study, we attempted such an analysis of chaperone-assisted protein folding in archaeal organisms through network approach using Picrophilus torridus as model system. The study revealed that DnaK protein of Hsp70 system acts as hub in protein-protein interaction network. However, DnaK protein was present only in a subset of archaeal organisms and absent from many archaea, especially members of Crenarchaeota phylum. Therefore, a similar network was created for another archaeal organism, Sulfolobus solfataricus, a member of Crenarchaeota. The chaperone network of S. solfataricus suggested that thermosomes played an integral part of hub proteins in archaeal organisms, where DnaK was absent. We further compared the chaperone network of archaea with that found in eukaryotic systems, by creating a similar network for Homo sapiens. In the human chaperone network, the UBC protein, a part of ubiquitination system, was the most important module, and interestingly, this system is known to be absent in archaeal organisms. Comprehensive comparison of these networks leads to several interesting conclusions regarding similarities and differences within archaeal chaperone machinery in comparison to humans.

Super-transient scaling in time-delay autonomous Boolean network motifs

Autonomous Boolean networks are commonly used to model the dynamics of gene regulatory networks and allow for the prediction of stable dynamical attractors. However, most models do not account for time delays along the network links and noise, which are crucial features of real biological systems. Concentrating on two paradigmatic motifs, the toggle switch and the repressilator, we develop an experimental testbed that explicitly includes both inter-node time delays and noise using digital logic elements on field-programmable gate arrays. We observe transients that last millions to billions of characteristic time scales and scale exponentially with the amount of time delays between nodes, a phenomenon known as super-transient scaling. We develop a hybrid model that includes time delays along network links and allows for stochastic variation in the delays. Using this model, we explain the observed super-transient scaling of both motifs and recreate the experimentally measured transient distributions.

In silico analysis of division times of Escherichia coli populations as a function of the partitioning scheme of non-functional proteins

Recent evidence suggests that cells employ functionally asymmetric partitioning schemes in division to cope with aging. We explore various schemes in silico, with a stochastic model of Escherichia coli that includes gene expression, non-functional proteins generation, aggregation and polar retention, and molecule partitioning in division. The model is implemented in SGNS2, which allows stochastic, multi-delayed reactions within hierarchical, transient, interlinked compartments. After setting parameter values of non-functional proteins’ generation and effects that reproduce realistic intracellular and population dynamics, we investigate how the spatial organization of non-functional proteins affects mean division times of cell populations in lineages and, thus, mean cell numbers over time. We find that division times decrease for increasingly asymmetric partitioning. Also, increasing the clustering of non-functional proteins decreases division times. Increasing the bias in polar segregation further decreases division times, particularly if the bias favors the older pole and aggregates’ polar retention is robust. Finally, we show that the non-energy consuming retention of inherited non-functional proteins at the older pole via nucleoid occlusion is a source of functional asymmetries and, thus, is advantageous. Our results suggest that the mechanisms of intracellular organization of non-functional proteins, including clustering and polar retention, affect the vitality of E. coli populations.

Noise analysis of genome-scale protein synthesis using a discrete computational model of translation

Noise in genetic networks has been the subject of extensive experimental and computational studies. However, very few of these studies have considered noise properties using mechanistic models that account for the discrete movement of ribosomes and RNA polymerases along their corresponding templates (messenger RNA (mRNA) and DNA). The large size of these systems, which scales with the number of genes, mRNA copies, codons per mRNA, and ribosomes, is responsible for some of the challenges. Additionally, one should be able to describe the dynamics of ribosome exchange between the free ribosome pool and those bound to mRNAs, as well as how mRNA species compete for ribosomes. We developed an efficient algorithm for stochastic simulations that addresses these issues and used it to study the contribution and trade-offs of noise to translation properties (rates, time delays, and rate-limiting steps). The algorithm scales linearly with the number of mRNA copies, which allowed us to study the importance of genome-scale competition between mRNAs for the same ribosomes. We determined that noise is minimized under conditions maximizing the specific synthesis rate. Moreover, sensitivity analysis of the stochastic system revealed the importance of the elongation rate in the resultant noise, whereas the translation initiation rate constant was more closely related to the average protein synthesis rate. We observed significant differences between our results and the noise properties of the most commonly used translation models. Overall, our studies demonstrate that the use of full mechanistic models is essential for the study of noise in translation and transcription.

Single-cell kinetics of a repressilator when implemented in a single-copy plasmid

Synthetic genetic clocks, such as the Elowitz-Leibler repressilator, will be key regulatory components of future synthetic circuits. We constructed a single-copy repressilator (SCR) by implementing the original repressilator circuit on a single-copy F-plasmid. After verifying its functionality, we studied its behaviour as a function of temperature and compared it with that of the original low-copy-number repressilator (LCR). Namely, we compared the period of oscillations, functionality (the fraction of cells exhibiting oscillations) and robustness to internal fluctuations (the fraction of expected oscillations that would occur). We found that, under optimal temperature conditions, the dynamics of the two systems differs significantly, although qualitatively they respond similarly to temperature changes. Exception to this is in the functionality, in which the SCR is higher at lower temperatures but lower at higher temperatures. Next, by adding IPTG to the medium at low and high concentrations during microscopy sessions, we showed that the functionality of the SCR is more robust to external perturbations, which indicates that the oscillatory behaviour of the LCR can be disrupted by affecting only a few of the copies in a cell. We conclude that the SCR, the first functional, synthetic, single-copy, ring-type genetic clock, is more robust to lower temperatures and to external perturbations than the original LCR. The SCR will be of use in future synthetic circuits, since it complements the array of tasks that the LCR can perform.

Effects of elongation delay in transcription dynamics

In the transcription process, elongation delay is induced by the movement of RNA polymerases (RNAP) along the DNA sequence, and can result in changes in the transcription dynamics. This paper studies the transcription dynamics that involved the elongation delay and effects of cell division and DNA replication. The stochastic process of gene expression is modeled with delay chemical master equation with periodic coefficients, and is studied numerically through the stochastic simulation algorithm with delay. We show that the average transcription level approaches to a periodic dynamics over cell cycles at homeostasis, and the elongation delay can reduce the transcription level and increase the transcription noise. Moreover, the transcription elongation can induce bimodal distribution of mRNA levels that can be measured by the techniques of flow cytometry.

Stochastic gene expression with delay

The expression of genes usually follows a two-step procedure. First, a gene (encoded in the genome) is transcribed resulting in a strand of (messenger) RNA. Afterwards, the RNA is translated into protein. We extend the classical stochastic jump model by adding delays (with arbitrary distributions) to transcription and translation. Already in the classical model, production of RNA and protein comes in bursts by activation and deactivation of the gene, resulting in a large variance of the number of RNA and proteins in equilibrium. We derive precise formulas for this second-order structure with the model including delay in equilibrium.

The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models

To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deducible from the mass action law and, in the limit n→∞, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability.

DDGni: dynamic delay gene-network inference from high-temporal data using gapped local alignment

MOTIVATION

Inferring gene-regulatory networks is very crucial in decoding various complex mechanisms in biological systems. Synthesis of a fully functional transcriptional factor/protein from DNA involves series of reactions, leading to a delay in gene regulation. The complexity increases with the dynamic delay induced by other small molecules involved in gene regulation, and noisy cellular environment. The dynamic delay in gene regulation is quite evident in high-temporal live cell lineage-imaging data. Although a number of gene-network-inference methods are proposed, most of them ignore the associated dynamic time delay.

RESULTS

Here, we propose DDGni (dynamic delay gene-network inference), a novel gene-network-inference algorithm based on the gapped local alignment of gene-expression profiles. The local alignment can detect short-term gene regulations, that are usually overlooked by traditional correlation and mutual Information based methods. DDGni uses 'gaps' to handle the dynamic delay and non-uniform sampling frequency in high-temporal data, like live cell imaging data. Our algorithm is evaluated on synthetic and yeast cell cycle data, and Caenorhabditis elegans live cell imaging data against other prominent methods. The area under the curve of our method is significantly higher when compared to other methods on all three datasets.

AVAILABILITY

The program, datasets and supplementary files are available at http://www.jjwanglab.org/DDGni/.

Stochastic modelling of biochemical systems of multi-step reactions using a simplified two-variable model

Background

A fundamental issue in systems biology is how to design simplified mathematical models for describing the dynamics of complex biochemical reaction systems. Among them, a key question is how to use simplified reactions to describe the chemical events of multi-step reactions that are ubiquitous in biochemistry and biophysics. To address this issue, a widely used approach in literature is to use one-step reaction to represent the multi-step chemical events. In recent years, a number of modelling methods have been designed to improve the accuracy of the one-step reaction method, including the use of reactions with time delay. However, our recent research results suggested that there are still deviations between the dynamics of delayed reactions and that of the multi-step reactions. Therefore, more sophisticated modelling methods are needed to accurately describe the complex biological systems in an efficient way.

Results

This work designs a two-variable model to simplify chemical events of multi-step reactions. In addition to the total molecule number of a species, we first introduce a new concept regarding the location of molecules in the multi-step reactions, which is the second variable to represent the system dynamics. Then we propose a simulation algorithm to compute the probability for the firing of the last step reaction in the multi-step events. This probability function is evaluated using a deterministic model of ordinary differential equations and a stochastic model in the framework of the stochastic simulation algorithm. The efficiency of the proposed two-variable model is demonstrated by the realization of mRNA degradation process based on the experimentally measured data.

Conclusions

Numerical results suggest that the proposed new two-variable model produces predictions that match the multi-step chemical reactions very well. The successful realization of the mRNA degradation dynamics indicates that the proposed method is a promising approach to reduce the complexity of biological systems.

Kinetics of gene expression in bacteria — From models to measurements, and back again

Dennis Salahub has contributed to several scientific areas. Likely, one of the lesser known contributions is the one on the study of the kinetics of gene expression in prokaryotes, which resulted in a delayed stochastic model of transcription and translation dynamics in bacteria. The model has become the basis for modeling stochastic genetic circuits and has provided a framework for interpreting recent measurements of transcripts production dynamics in live cells, one event at a time. Here, we review the contributions by Salahub and colleagues to the modeling strategies of gene expression as a multi-delayed stochastic process. Next, we describe recent findings, which build upon this work and provide experimental validation of the models. Finally, we discuss potential future developments in this field of research.

Chemical memory reactions induced bursting dynamics in gene expression

Memory is a ubiquitous phenomenon in biological systems in which the present system state is not entirely determined by the current conditions but also depends on the time evolutionary path of the system. Specifically, many memorial phenomena are characterized by chemical memory reactions that may fire under particular system conditions. These conditional chemical reactions contradict to the extant stochastic approaches for modeling chemical kinetics and have increasingly posed significant challenges to mathematical modeling and computer simulation. To tackle the challenge, I proposed a novel theory consisting of the memory chemical master equations and memory stochastic simulation algorithm. A stochastic model for single-gene expression was proposed to illustrate the key function of memory reactions in inducing bursting dynamics of gene expression that has been observed in experiments recently. The importance of memory reactions has been further validated by the stochastic model of the p53-MDM2 core module. Simulations showed that memory reactions is a major mechanism for realizing both sustained oscillations of p53 protein numbers in single cells and damped oscillations over a population of cells. These successful applications of the memory modeling framework suggested that this innovative theory is an effective and powerful tool to study memory process and conditional chemical reactions in a wide range of complex biological systems.

Effects of gene length on the dynamics of gene expression

In Escherichia coli, the nucleotide length of a gene is bound to affect its expression dynamics. From simulations of a stochastic model of gene expression at the nucleotide and codon levels we show that, within realistic parameter values, the nucleotide length affects RNA and protein mean levels, as well as the expected transient time for RNA and protein numbers to change, following a signal. Fluctuations in RNA and protein numbers are found to be minimized for a small range of lengths, which matches the means of the distributions of lengths found in E. coli of both essential and non-essential genes. The variance of the length distribution for essential genes is found to be smaller than for non-essential genes, implying that these distributions are far from random. Finally, gene lengths are shown to affect the kinetics of a genetic switch, namely, the correlation between temporal proteins numbers, the stability of the two noisy attractors of the switch, and how biased is the choice of noisy attractor. The stability increases with gene length due to increased 'memory' about the previous states of the switch. We argue that, by affecting the dynamics of gene expression and of genetic circuits, gene lengths are subject to selection.

Detecting sequence dependent transcriptional pauses from RNA and protein number time series

BACKGROUND

Evidence suggests that in prokaryotes sequence-dependent transcriptional pauses affect the dynamics of transcription and translation, as well as of small genetic circuits. So far, a few pause-prone sequences have been identified from in vitro measurements of transcription elongation kinetics.

RESULTS

Using a stochastic model of gene expression at the nucleotide and codon levels with realistic parameter values, we investigate three different but related questions and present statistical methods for their analysis. First, we show that information from in vivo RNA and protein temporal numbers is sufficient to discriminate between models with and without a pause site in their coding sequence. Second, we demonstrate that it is possible to separate a large variety of models from each other with pauses of various durations and locations in the template by means of a hierarchical clustering and a random forest classifier. Third, we introduce an approximate likelihood function that allows to estimate the location of a pause site.

CONCLUSIONS

This method can aid in detecting unknown pause-prone sequences from temporal measurements of RNA and protein numbers at a genome-wide scale and thus elucidate possible roles that these sequences play in the dynamics of genetic networks and phenotype.

Evolving kinetics of gene expression in stochastic environments

Recent studies have shown that the in vivo dynamics of RNA numbers in bacteria is regulated, to a great extent, by the kinetics of rate limiting steps in transcription. Strong evidence suggests that the kinetics of these steps is sequence dependent. We investigate the selective advantages of rate limiting steps of differing kinetics. For that, we model the kinetics of expression of a gene responsible for promoting cell division at the expense of resources in the environment in individual cells of a population. We model mutations that affect the kinetics of the rate limiting steps and selective pressure in various environmental conditions. Depletion of resources leads to cell death. We find that small changes in the evolutionary constraints can favor widely different noise levels in RNA and protein numbers. Increasing the cost in nutrients for division favors noisier expression. The results provide a better understanding of why different genes differ in the kinetics of production of RNA and proteins.

Modeling synthetic gene oscillators

Genetic oscillators have long held the fascination of experimental and theoretical synthetic biologists alike. From an experimental standpoint, the creation of synthetic gene oscillators represents a yardstick by which our ability to engineer synthetic gene circuits can be measured. For theorists, synthetic gene oscillators are a playground in which to test mathematical models for the dynamics of gene regulation. Historically, mathematical models of synthetic gene circuits have varied greatly. Often, the differences are determined by the level of biological detail included within each model, or which approximation scheme is used. In this review, we examine, in detail, how mathematical models of synthetic gene oscillators are derived and the biological processes that affect the dynamics of gene regulation.

Stability and multiattractor dynamics of a toggle switch based on a two-stage model of stochastic gene expression

A toggle switch consists of two genes that mutually repress each other. This regulatory motif is active during cell differentiation and is thought to act as a memory device, being able to choose and maintain cell fate decisions. Commonly, this switch has been modeled in a deterministic framework where transcription and translation are lumped together. In this description, bistability occurs for transcription factor cooperativity, whereas autoactivation leads to a tristable system with an additional undecided state. In this contribution, we study the stability and dynamics of a two-stage gene expression switch within a probabilistic framework inspired by the properties of the Pu/Gata toggle switch in myeloid progenitor cells. We focus on low mRNA numbers, high protein abundance, and monomeric transcription-factor binding. Contrary to the expectation from a deterministic description, this switch shows complex multiattractor dynamics without autoactivation and cooperativity. Most importantly, the four attractors of the system, which only emerge in a probabilistic two-stage description, can be identified with committed and primed states in cell differentiation. To begin, we study the dynamics of the system and infer the mechanisms that move the system between attractors using both the quasipotential and the probability flux of the system. Next, we show that the residence times of the system in one of the committed attractors are geometrically distributed. We derive an analytical expression for the parameter of the geometric distribution, therefore completely describing the statistics of the switching process and elucidate the influence of the system parameters on the residence time. Moreover, we find that the mean residence time increases linearly with the mean protein level. This scaling also holds for a one-stage scenario and for autoactivation. Finally, we study the implications of this distribution for the stability of a switch and discuss the influence of the stability on a specific cell differentiation mechanism. Our model explains lineage priming and proposes the need of either high protein numbers or long-term modifications such as chromatin remodeling to achieve stable cell fate decisions. Notably, we present a system with high protein abundance that nevertheless requires a probabilistic description to exhibit multistability, complex switching dynamics, and lineage priming.

Stochastic delay accelerates signaling in gene networks

The creation of protein from DNA is a dynamic process consisting of numerous reactions, such as transcription, translation and protein folding. Each of these reactions is further comprised of hundreds or thousands of sub-steps that must be completed before a protein is fully mature. Consequently, the time it takes to create a single protein depends on the number of steps in the reaction chain and the nature of each step. One way to account for these reactions in models of gene regulatory networks is to incorporate dynamical delay. However, the stochastic nature of the reactions necessary to produce protein leads to a waiting time that is randomly distributed. Here, we use queueing theory to examine the effects of such distributed delay on the propagation of information through transcriptionally regulated genetic networks. In an analytically tractable model we find that increasing the randomness in protein production delay can increase signaling speed in transcriptional networks. The effect is confirmed in stochastic simulations, and we demonstrate its impact in several common transcriptional motifs. In particular, we show that in feedforward loops signaling time and magnitude are significantly affected by distributed delay. In addition, delay has previously been shown to cause stable oscillations in circuits with negative feedback. We show that the period and the amplitude of the oscillations monotonically decrease as the variability of the delay time increases.

Delay in gene regulatory networks often arises from the numerous sequential reactions necessary to create fully functional protein from DNA. While the molecular mechanisms behind protein production and maturation are known, it is still unknown to what extent the resulting delay affects signaling in transcriptional networks. In contrast to previous studies that have examined the consequences of fixed delay in gene networks, here we investigate how the variability of the delay time influences the resulting dynamics. The exact distribution of “transcriptional delay” is still unknown, and most likely greatly depends on both intrinsic and extrinsic factors. Nevertheless, we are able to deduce specific effects of distributed delay on transcriptional signaling that are independent of the underlying distribution. We find that the time it takes for a gene encoding a transcription factor to signal its downstream target decreases as the delay variability increases. We use queueing theory to derive a simple relationship describing this result, and use stochastic simulations to confirm it. The consequences of distributed delay for several common transcriptional motifs are also discussed.

Probabilistic RNA partitioning generates transient increases in the normalized variance of RNA numbers in synchronized populations of Escherichia coli

We explore the effects of probabilistic RNA partitioning during cell division on the normalized variance of RNA numbers across generations of bacterial populations. We first characterize these effects in model cell populations, where gene expression is modeled as a delayed stochastic process, as a function of the synchrony in cell division, the rate of division, and the RNA degradation rate. We further explore the additional variance that arises if the partitioning is biased. Next, in Escherichia coli cells expressing RNA tagged with MS2d-GFP, we measured the normalized variance of RNA numbers across several generations, with cell divisions synchronized by heat shock. We show that synchronized cell populations exhibit transient increases in normalized variance following cell divisions, as predicted by the model, which are not observed in unsynchronized populations. We conclude that errors in partitioning of RNA molecules generate diversity between the offspring of individual bacteria and thus constitute a form of reproductive bet-hedging.

Automatic detection of changes in the dynamics of delayed stochastic gene networks and in vivo production of RNA molecules in Escherichia coli

MOTIVATION

Production and degradation of RNA and proteins are stochastic processes, difficulting the distinction between spurious fluctuations in their numbers and changes in the dynamics of a genetic circuit. An accurate method of change detection is key to analyze plasticity and robustness of stochastic genetic circuits.

RESULTS

We use automatic change point detection methods to detect non-spurious changes in the dynamics of delayed stochastic models of gene networks at run time. We test the methods in detecting changes in mean and noise of protein numbers, and in the switching frequency of a genetic switch. We also detect changes, following genes' silencing, in the dynamics of a model of the core gene regulatory network of Saccharomyces cerevisiae with 328 genes. Finally, from images, we determine when RNA molecules tagged with fluorescent proteins are first produced in Escherichia coli. Provided prior knowledge on the time scale of the changes, the methods detect them accurately and are robust to fluctuations in protein and RNA levels.

AVAILABILITY

Simulator: www.cs.tut.fi/~sanchesr/SGN/SGNSim.html

CONTACT

andre.ribeiro@tut.fi

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Dynamics of a genetic toggle switch at the nucleotide and codon levels

We study the dynamics of a model stochastic two-gene switch at the nucleotide and codon levels. First, we show that its stability, the mean lifetime of the noisy attractors, differs from that of a model where transcription and translation elongation are modeled as single-step delayed events, indicating the need of detailed models to study the dynamics of switches. Next, we vary the coupling between the two genes by varying the affinity of repressor proteins to the promoters and measure the mutual information between the two proteins times series. We find that there is a degree of coupling that maximizes information propagation between the two genes. This is explained by the effects of the coupling on mean and entropy of RNA and protein numbers of each gene, as well as correlation, 2-tuple entropy between the two proteins numbers, and, finally, the stability of the noisy attractors. We also find that increasing the rate of translation initiation increases the correlation between RNA and protein numbers and between the two proteins, due to increased stability of the noisy attractors. Increasing the rate of transcription or decreasing RNA degradation causes opposite effects to the correlation between RNA and proteins of each gene and the stability of the noisy attractors. Finally, we add a sequence-dependent transcription pause site and show that both its probability of occurrence, as well as its mean time length, affects the dynamics of the switch, further demonstrating the dependence of the dynamics of this circuit on sequence level events.

Cell-to-cell diversity in protein levels of a gene driven by a tetracycline inducible promoter

BACKGROUND

Gene expression in Escherichia coli is regulated by several mechanisms. We measured in single cells the expression level of a single copy gene coding for green fluorescent protein (GFP), integrated into the genome and driven by a tetracycline inducible promoter, for varying induction strengths. Also, we measured the transcriptional activity of a tetracycline inducible promoter controlling the transcription of a RNA with 96 binding sites for MS2-GFP.

RESULTS

The distribution of GFP levels in single cells is found to change significantly as induction reaches high levels, causing the Fano factor of the cells' protein levels to increase with mean level, beyond what would be expected from a Poisson-like process of RNA transcription. In agreement, the Fano factor of the cells' number of RNA molecules target for MS2-GFP follows a similar trend. The results provide evidence that the dynamics of the promoter complex formation, namely, the variability in its duration from one transcription event to the next, explains the change in the distribution of expression levels in the cell population with induction strength.

CONCLUSIONS

The results suggest that the open complex formation of the tetracycline inducible promoter, in the regime of strong induction, affects significantly the dynamics of RNA production due to the variability of its duration from one event to the next.

Stochastic sequence-level model of coupled transcription and translation in prokaryotes

Background

In prokaryotes, transcription and translation are dynamically coupled, as the latter starts before the former is complete. Also, from one transcript, several translation events occur in parallel. To study how events in transcription elongation affect translation elongation and fluctuations in protein levels, we propose a delayed stochastic model of prokaryotic transcription and translation at the nucleotide and codon level that includes the promoter open complex formation and alternative pathways to elongation, namely pausing, arrests, editing, pyrophosphorolysis, RNA polymerase traffic, and premature termination. Stepwise translation can start after the ribosome binding site is formed and accounts for variable codon translation rates, ribosome traffic, back-translocation, drop-off, and trans-translation.

Results

First, we show that the model accurately matches measurements of sequence-dependent translation elongation dynamics. Next, we characterize the degree of coupling between fluctuations in RNA and protein levels, and its dependence on the rates of transcription and translation initiation. Finally, modeling sequence-specific transcriptional pauses, we find that these affect protein noise levels.

Conclusions

For parameter values within realistic intervals, transcription and translation are found to be tightly coupled in Escherichia coli, as the noise in protein levels is mostly determined by the underlying noise in RNA levels. Sequence-dependent events in transcription elongation, e.g. pauses, are found to cause tangible effects in the degree of fluctuations in protein levels.

Inference of kinetic parameters of delayed stochastic models of gene expression using a markov chain approximation

We propose a Markov chain approximation of the delayed stochastic simulation algorithm to infer properties of the mechanisms in prokaryote transcription from the dynamics of RNA levels. We model transcription using the delayed stochastic modelling strategy and realistic parameter values for rate of transcription initiation and RNA degradation. From the model, we generate time series of RNA levels at the single molecule level, from which we use the method to infer the duration of the promoter open complex formation. This is found to be possible even when adding external Gaussian noise to the RNA levels.

Information propagation within the Genetic Network of Saccharomyces cerevisiae

Background

A gene network's capacity to process information, so as to bind past events to future actions, depends on its structure and logic. From previous and new microarray measurements in Saccharomyces cerevisiae following gene deletions and overexpressions, we identify a core gene regulatory network (GRN) of functional interactions between 328 genes and the transfer functions of each gene. Inferred connections are verified by gene enrichment.

Results

We find that this core network has a generalized clustering coefficient that is much higher than chance. The inferred Boolean transfer functions have a mean p-bias of 0.41, and thus similar amounts of activation and repression interactions. However, the distribution of p-biases differs significantly from what is expected by chance that, along with the high mean connectivity, is found to cause the core GRN of S. cerevisiae's to have an overall sensitivity similar to critical Boolean networks. In agreement, we find that the amount of information propagated between nodes in finite time series is much higher in the inferred core GRN of S. cerevisiae than what is expected by chance.

Conclusions

We suggest that S. cerevisiae is likely to have evolved a core GRN with enhanced information propagation among its genes.

Expanding the modern synthesis

The Modern Evolutionary Synthesis formalizes the role of variation, heredity, differential reproduction and mutation in population genetics. Here we explore a mathematical structure, based on the asymptotic limit theorems of communication theory, that instantiates the punctuated dynamic relations of organisms with their embedding environments, including the possibility of the transfer of heritage information between different classes of organism. The approach applies a standard coevolutionary argument to genes, environment, and gene expression reconfigured as interacting information sources. In essence, we provide something of a formal roadmap for the modernization of the Modern Synthesis, making applications to both relatively rapid evolutionary punctuated equilibrium and to the conservation of ecological interactions across deep evolutionary time.

Dynamical effects of transcriptional pause-prone sites

We study how long pause-prone sites, commonly sequence-dependent, affect transcription and RNA temporal levels in a delayed stochastic model of transcription at the single nucleotide level. We vary pause propensity, duration and the probability of premature termination of elongation at the pause site. We also study the effects of multiple pause sites. We show that pause sites can be used to fine-tune noise strength and burst size distribution of RNA levels. Varying pause rate and duration alone affects bursting but noise is not significantly affected. Noise strength can be changed by varying both parameters and, even more pronouncedly, by varying the probability of premature termination. Adding multiple pause sites amplifies the increase in noise and bursting. This regulatory mechanism of noise and bursting, being evolvable, may partially explain how different genes exhibit a wide spectrum of different behaviors. The results might assist the engineering of genes with a desired degree of noise.

Effects of transcriptional pausing on gene expression dynamics

Stochasticity in gene expression affects many cellular processes and is a source of phenotypic diversity between genetically identical individuals. Events in elongation, particularly RNA polymerase pausing, are a source of this noise. Since the rate and duration of pausing are sequence-dependent, this regulatory mechanism of transcriptional dynamics is evolvable. The dependency of pause propensity on regulatory molecules makes pausing a response mechanism to external stress. Using a delayed stochastic model of bacterial transcription at the single nucleotide level that includes the promoter open complex formation, pausing, arrest, misincorporation and editing, pyrophosphorolysis, and premature termination, we investigate how RNA polymerase pausing affects a gene's transcriptional dynamics and gene networks. We show that pauses' duration and rate of occurrence affect the bursting in RNA production, transcriptional and translational noise, and the transient to reach mean RNA and protein levels. In a genetic repressilator, increasing the pausing rate and the duration of pausing events increases the period length but does not affect the robustness of the periodicity. We conclude that RNA polymerase pausing might be an important evolvable feature of genetic networks.

Effects of the promoter open complex formation on gene expression dynamics

Little is known about the biological mechanisms that shape the distribution of intervals between the completion of RNA molecules (T(p)RNA) , and thus transcriptional noise. We characterize numerically and analytically how the promoter open complex delay (tau(P)) and the transcription initiation rate (k(t)) shape T(p)RNA. From this, we assess the noise and mean of transcript levels and show that these can be tuned both independently and simultaneously by tau(P) and k(t). Finally, we characterize how tau(P) affects bursting in RNA production and show that the tau(P) measured for a lac promoter best fits independent measurements of the burst distribution of the same promoter. Since tau(P) affects noise in gene expression, and given that it is sequence dependent, it is likely to be evolvable.